The relevant passage concerning this is the following:
"While the words 'designates,' 'satisfies,' and 'defines' express
relations (between certain expressions and the objects 'referred to'
by these expressions), the word 'true' is of a different logical
nature: it expresses a property (or denotes a class) of certain
expressions, viz, of sentences. However, it is easily seen that all
the formulations which were given earlier and which aimed to explain
the meaning of this word ... referred not only to sentences
themselves, but also to objects 'talked about' by these sentences, or
possibly to 'states of affairs' described by them. And, moreover, it
turns out that the simplest and the most natural way of obtaining an
exact definition of truth is one which involves the use of other
semantic notions, e.g., the notion of satisfaction. It is for these
reasons that we count the concept of truth which is discussed here
among the concepts of semantics, and the problem of defining truth
proves to be closely related to the more general problem of setting up
the foundations of theoretical semantics." (Alfred Tarski, The
Semantic Conception of Truth)
An example which Tarski gives to illustrate the nature of the notion of satisfaction is the following:
"The sentence 'snow is white' is true if, and only if, snow is white"
Satisfaction involves a relation between a sentence and the sentence's referent, or as Tarski puts it, the "objects 'talked about' by these sentences," and this relation is unavoidable. Independently of whatever logical formulation is used, i.e. whether it amounts to a finite lists of sentences, whether it involves sentential logic or quantification; regardless of the formulation, the most basic elements of any metalanguage will always consist of atomic formulas whose truth value depends on this relation. The reason for this is that it is essential to what language is.
On the one hand, there is reality whose nature and existence is independent from our knowing anything about it, and on the other hand, there is the various means of representing such reality in such a way that it is conducive to understanding and communication. Language is one such means, and it's very reason for being is to serve in this representative capacity, so without the relation which underlies the notion of satisfaction, language is nothing at all. It makes no sense to speak of the truth of a sentence by itself independent of what it represents, because it would be reduced to a useless sequence of symbols. In the same way, it doesn't make sense to speak of reality as true independent of any perception, description, understanding or representation, because truth by its very nature is the aptitude of such representations to accurately portray reality. This is precisely what Tarski was saying when he said that truth "expresses a property (or denotes a class) of certain expressions, viz, of sentences."
Jamin Asay writes:
"Tarski’s interest was never in replacing our ordinary conception of
truth with the kind of definitions he offers. Rather, Tarski’s
definitions work in conjunction with our ordinary conception of truth.
We know that Tarski’s definitions are successful only if they are
materially adequate, in which case they entail all the T-sentences.
For the T-sentences to provide an independent check on Tarski’s
definitions, they must be expressing important facts about our
ordinary conception of truth (or, at least, about the truth conditions
of our sentences), and not vacuous logical truths." (Jamin Asay,
"Tarski and Primitivism About Truth")
Therefore, the notion of satisfaction is essential to any definition of truth because it refers to the aptitude of sentences to fulfill their essential purpose. And no, truth cannot be defined as the logical product of all instances of a schema without resorting to the concept of satisfaction, because the notion of satisfaction is required to compose any such schema at the atomic level.